A129710: Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

Matches ?

• A129710 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?

Row sums ? A185357, A132636, A212804, A077373, A000045, A202278, A274163, A248740

Submission text ?

Triangle begins:
  0: 1;
  1: 2;
  2: 2,  1;
  3: 2,  3;
  4: 2,  5,  1;
  5: 2,  7,  4;
  6: 2,  9,  9,  1;
  7: 2, 11, 16,  5;
  8: 2, 13, 25, 14, 1;
  9: 2, 15, 36, 30, 6;

The row polynomials satisfy P(n,x) = P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
  0: 1;
  1: 2;
  2: 2,  1;
  3: 2,  3;
  4: 2,  5,  1;
  5: 2,  7,  4;
  6: 2,  9,  9,  1;
  7: 2, 11, 16,  5;
  8: 2, 13, 25, 14, 1;
  9: 2, 15, 36, 30, 6;

The row polynomials satisfy P(n,x) = P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
  by Liu–Wang theorem for three-term recurrences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Liu–Wang theorem for three-term recurrences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Liu–Wang theorem for three-term recurrences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
  by Liu–Wang theorem for three-term recurrences
• Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a real-rooted image with every root <= -c/b = 0
  by Constant-current basis map
• Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a PF image (zero allowed)
  by Constant-current basis map

Proof chains ? JSON

• Constant-current basis map [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ? step › direct check
• Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 10

• basis_map_A008277 of Triangle read by rows: T(n,k) is the number of Fibonacci b… Image under a basis map u^k -> P_k (A008277)
• basis_map_A013611 of Triangle read by rows: T(n,k) is the number of Fibonacci b… Image under a basis map u^k -> P_k (A013611)
• basis_map_A013609 of Triangle read by rows: T(n,k) is the number of Fibonacci b… Image under a basis map u^k -> P_k (A013609)
• Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ..… Image under a basis map u^k -> P_k (A007318)
• basis_map_A013610 of Triangle read by rows: T(n,k) is the number of Fibonacci b… Image under a basis map u^k -> P_k (A013610)
• one_plus_derivative of Triangle read by rows: T(n,k) is the number of Fibonacci… Image under a real-rootedness preserver (derivative polynomial, shift 0)
• derivative of Triangle read by rows: T(n,k) is the number of Fibonacci binary w… Image under a real-rootedness preserver (derivative polynomial, shift 0)
• theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of Fibona… Image under a real-rootedness preserver (euler polynomial, shift 0)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003